closedBall_rpow_sub_one_eq_empty_aux

English

For all real b,x, log x / log b = log_b(b,x).

Русский

Для всех вещественных b,x выполнено log x / log b = log_b b x.

LaTeX

$$$$\dfrac{\log x}{\log b} = \log_b b x.$$$$

Lean4

theorem closedBall_rpow_sub_one_eq_empty_aux {r t : ℝ} (hr : 0 < r) (ht : 1 < t) :
    Metric.closedBall (0 : E) (t ^ (-r⁻¹) - 1) = ∅ :=
  by
  rw [Metric.closedBall_eq_empty, sub_neg]
  exact Real.rpow_lt_one_of_one_lt_of_neg ht (by simp only [hr, Right.neg_neg_iff, inv_pos])

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